= Opaque barrier approximation
{title2=$T\sim\frac{16E(V_0-E)}{V_0^2}e^{-2\kappa a}$}
For fixed $0<E<V_0$ and $\kappa a\gg1$, the exact <rectangular barrier transmission probability> gives
$$
T\sim\frac{16E(V_0-E)}{V_0^2}\exp\!\left[-\frac{2a\sqrt{2m(V_0-E)}}{\hbar}\right].
$$
This is the leading term of an <asymptotic expansion>: the <wavefunction> amplitude scale is $e^{-\kappa a}$, and the <transmission probability> scale is $e^{-2\kappa a}$, with an interface prefactor. Increasing width by $\Delta a$ multiplies this approximation by $e^{-2\kappa\Delta a}$. At fixed incident <energy>, increasing barrier height increases the <tunnelling decay constant> as $\sqrt{V_0-E}$; heavier <nonrelativistic particles> also have stronger suppression. For the numerical example in <rectangular barrier transmission probability>, the approximation gives $0.01397$, compared with the exact $0.01388$.
The regime condition matters. Near $E=V_0$ at fixed width, $\kappa a$ tends to zero and the <opaque barrier approximation> fails; use the exact threshold value. For $\kappa a\ll1$, expansion of the <hyperbolic sine> gives
$$
T\simeq\left[1+\frac{mV_0^2a^2}{2\hbar^2E}\right]^{-1}.
$$
This tends to one as $a\to0$ at fixed $E,V_0$, but the condition $\kappa a\ll1$ alone does not imply nearly perfect transmission if the remaining factor is large. At $E>V_0$, use the oscillatory formula and its <scattering resonances>, rather than an <exponential decay> law.
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